We study spherical quadrilaterals whose angles are odd
multiples of $\pi/2$, and the equivalent accessory parameter problem
for the Heun equation. We obtain a classification of these
quadrilaterals up to isometry. For given angles, there are finitely
many one-dimensional continuous families which we enumerate. In each
family the conformal modulus is either bounded from above or bounded
from below, but not both, and the numbers of families of these two
types are equal. The results can be translated to classification of
Heun’s equations with real parameters, whose exponent differences
are odd multiples of $1/2$, with unitary monodromy.

A (marked) circular polygon is a closed disk $Q$ with marked
boundary points $a_{0},\ldots,a_{n-1}$, which are called corners, enumerated cyclically according to the positive
orientation of $\partial Q$, equipped with a Riemannian metric of
constant curvature 1 with conic singularities at the corners, and
such that each side $(a_{j},a_{j+1})$ has constant geodesic
curvature. Two such polygons $Q$ and $Q^{\prime}$ are congruent if
there is an orientation-preserving isometry between them which sends
each corner $a_{j}$ of $Q$ to the corner $a_{j}^{\prime}$ of $Q^{\prime}$.

Polygons with $n=2,\ 3$ and 4 are called digons, triangles
and quadrilaterals, respectively.

If each side has zero geodesic curvature then $Q$ is called a spherical polygon. At every corner $a_{j}$, an interior angle
$\alpha_{j}\geq 0$ is defined, and in what follows we measure all
angles in half-turns. So angle $\alpha$ means an angle of
$\pi\alpha$ radians, in particular, “integer angle” is an integer
multiple of $\pi$ radians. A circular quadrilateral $Q$ whose angles
are odd multiples of $1/2$ is called a circular rectangle. If
such a quadrilateral is spherical, it is called a spherical
rectangle. In this paper we describe the set of spherical
rectangles with prescribed angles.

As every surface of positive curvature 1 is locally isometric to a
piece of the unit sphere, every circular polygon can be described in
terms of the developing map $f:Q\to\mathbf{\overline{C}}$ which is an analytic
function on $Q\setminus\{a_{0},\ldots,a_{n-1}\}$ mapping every side
into a circle on the Riemann sphere. For spherical polygons these
circles are geodesic (great circles). This function $f$ is a local
homeomorphism at each point except the corners, and at a corner $a$
satisfies

$\displaystyle f(z)-f(a)\sim c(z-a)^{\alpha},$

where $\alpha>0$ is the angle at this corner. (If $\alpha=0$, the
right-hand side has to be replaced by $c/\log(z-a)$.)

Each such function defines a circular polygon by the pull-back of
the spherical metric from $\mathbf{\overline{C}}$ to $Q$. If none of the $\alpha_{j}$
equals 1, then the pair $(f,a_{0})$ defines the polygon uniquely.
Two pairs $(f_{1},a_{0})$ and $(f_{2},a_{0}^{\prime})$ define congruent
polygons if $f_{2}=\psi\circ f_{1}\circ\phi$, where $\psi$ is a rotation
of the Riemann sphere, and $\phi$ a conformal automorphism of the
disk with the property $\phi(a_{j})=a_{j}^{\prime}$ for all $j$.

This paper is a part of the project whose goal is to understand
metrics of constant positive curvature with conic singularities on
compact surfaces, see [Troyanov1991], [Luo and Tian1992], [Eremenko2004], [Eremenko et al.2014]; [Eremenko et al.2016a]; [Eremenko et al.2016b], [Mondello and Panov2016], [Lin and Wang2010], [Chai et al.2015]. An
important class of such metrics can be obtained by gluing a
spherical polygon to its mirror image isometrically along the
boundary. Metrics of positive curvature on the sphere obtained in
this way are characterized by the symmetry property: all conic
singularities belong to a circle on the sphere, and the metric is
symmetric with respect to this circle.

In this paper, we classify spherical rectangles. The cases when at
least one of the angles of a spherical quadrilateral is integer were
considered in [Eremenko et al.2006]; [Eremenko et al.2014]; [Eremenko et al.2016a]; [Eremenko et al.2016b].

If we use the upper half-plane as $Q$, then the developing map of
every circular quadrilateral is a ratio of two linearly independent
solutions of the Heun equation

with the standard normalization $(a_{0},a_{1},a_{2},a_{3})=(0,1,a,\infty)$,
$a\in(1,+\infty)$. Here $\alpha_{j}$ are the angles at the corners,
and $\lambda$ is a real accessory parameter. Each pair of
linearly independent solutions of such an equation defines a
circular quadrilateral. Different pairs of linearly independent
solutions of the same equation define equivalent
quadrilaterals: their developing maps are related by
post-composition with linear-fractional transformations. We will
later see that an equivalence class may contain at most one
spherical quadrilateral, up to congruence.

The condition that an equivalence class contains a spherical
quadrilateral translates to the following condition on the Heun
equation: the projective monodromy group must be conjugate to a
subgroup of $SU(2)$. So the problem of classification of spherical
quadrilaterals with prescribed corners and angles is equivalent to
the problem of classification of Heun’s equation with prescribed
$a_{j}$ and $\alpha_{j}$ whose monodromy is unitarizable, that is
conjugate to a subgroup of $SU(2)$. The correspondence between the
metrics on the sphere and Heun’s equations with unitarizable
monodromy is bijective. Each symmetric metric on the sphere
corresponds to two spherical quadrilaterals which are related by an
anti-conformal involution, and to a unique normalized Heun equation
with real $a,\lambda$ and unitarizable monodromy.

In the case that all angles are odd multiples of $1/2$, Heun’s
equation can be explicitly solved in terms of elliptic integrals.
This fact was discovered by [Darboux1882] who generalized
Hermite’s work [Hermite1912] for the Lamé equation. For the study
of general circular rectangles in connection with Heun’s equation
with $\alpha_{j}$ multiples of $1/2$ we refer to the paper of [Van Vleck1902].

We recall how this explicit solution is obtained.

Theorem A.

Suppose that all $\alpha_{j}$ in (1.1)–(1.2) are odd
multiples of $1/2$. Then there are two linearly independent
solutions of (1.1) whose ratio is of the form

where $p$ and $q$ are the coefficients in front of $y^{\prime}$ and $y$ in
(1.1).

Equation (1.5) has one-dimensional space of polynomial
solutions of degree

$\displaystyle\deg P=\sum_{j=0}^{3}\alpha_{j}-2.$

(1.6)

This permits to find $P$ by rational operations. That $P$ satisfies
(1.5) guarantees that all residues of the integrand in
(1.4) are of the form $\pm c$ with some real $c$. The
condition $c=1$ defines $P$ up to a sign.

Periods of the integral (1.4), other than those coming
from the residues, form a lattice generated by two canonical periods
(integrals over adjacent real segments). Of these two canonical
periods one is real and another is pure imaginary. The condition
that the monodromy of (1.1) is unitarizable means that both
periods must be imaginary, therefore the real period must vanish.
For a fixed real $a$, and given angles, this imposes a
transcendental equation on $\lambda$. It is not clear how to
determine or estimate the number of real solutions of this equation,
but for small angles $\alpha_{k}$ it can be solved numerically. The
results of computation are described in Sect. 4.

Instead we use a geometric method which allows us to classify
spherical rectangles, and describe their geometry. The following
elementary statement was proved in [Eremenko and Gabrielov2015].

Proposition 1.1.

Let $f$ be the developing map of a spherical rectangle $Q$. Then
there are two opposite sides of $Q$ whose $f$-images are contained
in the same circle, and the other pair of opposite sides is mapped
to distinct circles.

Thus the boundary of a spherical rectangle $Q$ is mapped by $f$ to
the union of three great circles, one of them, say $C$, being
orthogonal to the other two, $C^{\prime}$ and $C^{\prime\prime}$. Let $\theta\in(0,1)$ be
the angle between the circles $C^{\prime}$ and $C^{\prime\prime}$. There are two choices
for this angle (the other one being $1-\theta$). See Definition
3.4 below for the unique choice of the angle $\theta$
associated with a spherical rectangle $Q$.

The $f$-preimage in $Q$ of the three circles is called the net
of $Q$. The net is a combinatorial invariant of $Q$, defined up to
an orientation-preserving homeomorphism of $Q$ respecting its
initial corner $a_{0}$.

Proposition 1.2.

A marked spherical rectangle $Q$ is defined uniquely up to isometry
by its net and the angle $\theta\in(0,1)$.

This will be proved in Sect. 3. In Sect. 3, we
will explicitly describe all possible nets of spherical rectangles
(Theorems 3.1 and 3.5). As a
consequence we will obtain the following necessary and sufficient
conditions on the angles of a spherical rectangle:

Theorem 1.3.

Let $A_{0},\ldots,A_{3}$ be non-negative integers, and

$\displaystyle\delta=(A_{1}+A_{3}-A_{0}-A_{2})/2.$

(1.7)

Then for the existence of a spherical rectangle with angles
$\alpha_{j}=A_{j}+1/2$ it is necessary and sufficient that one of the
following conditions be satisfied: either

$\displaystyle\delta\geq 1,\quad A_{1}\geq 1,\quad A_{3}\geq 1,$

(1.8)

or

$\displaystyle\delta\leq-1,\quad A_{0}\geq 1,\quad A_{2}\geq 1.$

(1.9)

To state our next result we need some definitions. We recall that we
consider marked spherical rectangles. Two of them are
congruent if there is an orientation-preserving isometry of $Q$
respecting the initial corner $a_{0}$.

Every quadrilateral can be mapped conformally onto a flat rectangle
with vertices $(0,1,1+iK,iK)$ and all angles $1/2$, such that $a_{0}$
maps to 0. The number $K$ is called the modulus of the
quadrilateral.

Each pair $(\Gamma,\theta)$, where $\Gamma$ is a net and
$\theta\in(0,1)$, defines a marked spherical rectangle
$Q(\Gamma,\theta)$ (see Theorem 3.5). Thus the set of
all spherical rectangles with given angles consists of curves
$\theta\mapsto Q(\Gamma,\theta)$ parameterized by $\theta$ and
labeled by the nets. The modulus $K$ of $Q(\Gamma,\theta)$ is a
continuous function of $\theta$. There are two kinds of these
curves:

On the curves of the first kind, $K\to 0$ as $\theta\to 0$, while
$K$ tends to a non-zero limit $K_{\mathrm{crit}}(\Gamma)$ as
$\theta\to 1$.

On the curves of the second kind, $K\to+\infty$ as $\theta\to 0$,
while $K$ tends to a non-zero limit $K_{\mathrm{crit}}(\Gamma)$ as
$\theta\to 1$.

This is proved in Theorem 4.2, and we give few examples of
computation of the limits $K_{\mathrm{crit}}$ in Sect.
4. In all our examples $K$ is a monotone function of
$\theta$. This is proved in [Eremenko and Gabrielov2015] for the simplest family of
spherical quadrilaterals with angles $(3/2,1/2,3/2,1/2)$ but it is
unlikely that this property holds in general. However, it is true
for sufficiently small and large values of $K$.

Proposition 1.4.

Each curve $Q(\Gamma,\theta)$ has finitely many intervals on which
$K$ is monotone. In particular, for sufficiently small (resp.,
large) $K>0$, there is a unique spherical rectangle with the modulus
$K$ in a curve $Q(\Gamma,\theta)$ of the first (resp., second)
kind.

This follows from the general theory of o-minimal structures (see,
e.g., [van den Dries1998]), since the integral in (1.4) is a
Pfaffian function in the sense of [Khovanskii1980]. It was
shown in [Speissegger1999] (see also [Wilkie1999]) that the structure
generated by Pfaffian functions is o-minimal.

Our final results count the nets for spherical rectangles with given
angles.

The quadruple $(A_{0},\dots,A_{3})$ is special if $\delta$ in
(1.7) is an odd integer and one of the following holds:
either $A_{1}\geq\delta>0$ and $A_{3}\geq\delta>0$ or $A_{0}\geq-\delta>0$
and $A_{2}\geq-\delta>0$.

Note that conditions (1.8) and (1.9) are satisfied when
$M_{1}>0$ and $M_{2}>0$, respectively.

Theorem 1.5.

For a special quadruple $(A_{0},\ldots,A_{3})$ satisfying either
(1.8) or (1.9) there exist $2N$ one-parametric families of
congruence classes of marked spherical rectangles with angles
$A_{j}+1/2$. If $(A_{0},\ldots,A_{3})$ is not special but satisfies
(1.8) (resp., (1.9)) then there exist $2M_{1}$ (resp., $2M_{2}$)
one-parametric families of congruence classes of marked spherical
rectangles with angles $A_{j}+1/2$.

Each family is parameterized by $\theta\in(0,1)$ (see Definition
3.4). Each family contains either rectangles of
arbitrarily small moduli or arbitrarily large moduli but not both.
The numbers of families of both types are equal, so for each type
this number is either $N$ or $M_{1}$ or $M_{2}$, depending on
$(A_{0},\dots,A_{3})$.

Remark 1.6.

Theorem 1.5 and Proposition 1.4 imply
that the number of spherical rectangles with given angles $A_{j}+1/2$
is exactly $N$ or $M_{1}$ or $M_{2}$, depending on $(A_{0},\dots,A_{3})$,
for sufficiently small and large values of $K$.

For fixed angles $\alpha_{j}\in\mathbf{N}^{+}+1/2$, consider the two-parametric
family of Heun’s equations (1.1) with parameters
$(a,\lambda)\in(1,+\infty)\times\mathbf{R}.$ Equivalence classes of circular
rectangles are in correspondence to such Heun’s equations.
One-parametric families of spherical rectangles of Theorem
1.5 correspond to smooth disjoint curves in the
half-plane $(a,\lambda)$, each having one end in this half-plane. On
the other end, either $a\to 1$ or $a\to\infty$.

When $\theta=p/q$ is rational, the monodromy group of the developing
map is finite, so $f$ is an algebraic function. In this case,
$f=g^{-1}\circ h$, where

$\displaystyle g=-\frac{1}{4}\left(z^{q}+\frac{1}{z^{q}}-2\right),$

and $h$ is a rational Belyi function, which means that the only
critical values of $h$ are $0,1,\infty$, [Schneps1994]. Function
$g$ is also a Belyi function, it is called the fundamental rational
function of the dihedral group [Klein1993]. The set $g^{-1}(\mathbf{R})$
consists of the unit circle and $q$ lines $\{z=t\exp(\pi ik/q):t\in\mathbf{R},\;k=0,\ldots,q-1\}.$ In the simplest case $\theta=1/2$, the
image $f(\partial Q)$ is contained in the union of the unit circle
$C$, real line $C^{\prime}$, and imaginary line $C^{\prime\prime}$. The monodromy is the
Klein Viergroup $\mathbf{Z}^{2}\times\mathbf{Z}^{2}$, represented as $\{z,-z,1/z,-1/z\}$ and $g$ has the property that $g^{-1}(\mathbf{R})=C\cup C^{\prime}\cup C^{\prime\prime}$. Then it is easy to see that our net, together with its
reflection in the real line, coincides with $h^{-1}(\mathbf{R})$. The set
$h^{-1}(\mathbf{R})$ for a Belyi function $h$ is a triangulation of the
sphere with all vertices of even degree. Therefore our
classification of the nets can be restated as classification of
triangulations ${\mathbf{T}}$ of the sphere with the following
properties:

(a)

${\mathbf{T}}$ is symmetric with respect to $\mathbf{R}$, and $\mathbf{R}$
is contained in the 1-skeleton of ${\mathbf{T}}$,

(b)

There are four vertices $a_{j}$ of ${\mathbf{T}}$ on the
real line of prescribed orders $4A_{j}+2$.

(c)

All other vertices of ${\mathbf{T}}$ have order 4.

For $q\in\{2,3\}$ and $(A_{0},\ldots,A_{3})=(1,0,1,0)$, algebraic
developing maps are explicitly written in [Eremenko and Gabrielov2015].

In this section we prove a preliminary result for classification of
nets. Roughly speaking it says that every spherical rectangle is a
union of two spherical triangles.

As we prove this result by induction, it is convenient to consider a
more general class of spherical polygons, characterized by the
property that the developing map sends their sides to three
transversally intersecting great circles and corners to the
intersection points of these circles. The net $\Gamma$ defines a
triangulation of such a polygon $Q$, each face of it being mapped by
$f$ one-to-one onto one of the triangles into which the three
circles partition the sphere. This triangulation satisfies the
following properties:

(P1)

Each vertex inside $Q$ has degree 4;

(P2)

All boundary vertices, other than corners of $Q$, have
degree 3.

Combining this triangulation with its mirror copy, we obtain a
triangulation ${\mathbf{T}}$ of the sphere satisfying the following
properties:

(S1)

${\mathbf{T}}$ is symmetric with respect to a circle $S$
contained in the 1-skeleton of ${\mathbf{T}}$;

(S2)

Each vertex of ${\mathbf{T}}$ has even degree, and all
its vertices not contained in $S$ have degree 4.

It is easy to show that the nets of spherical polygons with all
sides mapped to three transversal circles and all corners to
intersection points of those circles are in one-to-one
correspondence with triangulations of the sphere satisfying (S1) and
(S2).

Two nets are combinatorially equivalent if they can be obtained from
each other by an orientation-preserving homeomorphism (mapping
corners to corners and sides to sides) preserving the initial
corner.

If $C$ is any of the three circles, its preimage in $Q$ is called
$C$-net, denoted $\Gamma_{C}$. An arc of the net
$\Gamma_{C}$ (or an arc of $\Gamma$ if $C$ is not specified) is a
connected component of $\Gamma_{C}\setminus\partial Q$. Since $f$ is a
local homeomorphism on the interior of $Q$, an arc may be
homeomorphic to either an open interval with both ends on the
boundary of $Q$ (possibly, at the same corner of $Q$) or a circle in
the interior of $Q$. We’ll show below (see Corollary 2.3)
that an arc of a spherical rectangle $Q$ must have at least one end
at a corner of $Q$. In particular, an arc of a spherical rectangle
cannot be a circle. An arc is called short if it does not
intersect other arcs of $\Gamma$. Any two arcs of the same net
$\Gamma_{C}$ are disjoint.

Definition 2.1.

We say that a spherical polygon $Q$ is reducible if its net
has an arc with endpoints at two distinct corners of $Q$. Such an
arc partitions $Q$ into two proper sub-polygons. Otherwise, $Q$ is
irreducible. We say that $Q$ is primitive if it is
irreducible and its net does not contain an arc with both ends at
the same corner of $Q$.

Theorem 2.2.

Let $Q$ be a spherical $n$-gon such that all its sides are mapped to
three transversal great circles by the developing map, and all its
corners are mapped to intersection points of those circles. Then
either $n\leq 3$ or there is a triangulation of $Q$ by $n-3$ disjoint
arcs of its net, each of them connecting two non-adjacent corners of $Q$.

Proof.

It is enough to show that, unless $Q$ is a digon or a triangle,
there exists an arc of its net $\Gamma$ connecting two of its
non-adjacent corners. We prove this by induction on the number $N$
of faces of $\Gamma$. If $\Gamma$ has one face then, since any face
of $\Gamma$ is a triangle, $Q$ is a triangle.

If $N>1$ then there exists an arc $\gamma$ of $\Gamma$ adjacent to a
point $p$ on the boundary of $Q$. Otherwise the face of $\Gamma$
adjacent to its boundary would not be simply connected.

If $\gamma$ connects two distinct corners $p$ and $q$ of $Q$ then
either $p$ and $q$ are non-adjacent and we are done, or $p$ and $q$
are adjacent corners of $Q$, and $\gamma$ partitions $Q$ into a
digon and a polygon $Q^{\prime}$ with the same number of corners as $Q$ and
a smaller than $N$ number of faces of its net. By inductive
hypothesis, unless $Q^{\prime}$ (and thus $Q$) is a digon or a triangle,
there is an arc $\gamma^{\prime}$ of $Q^{\prime}$ connecting two of its non-adjacent
corners. In the latter case, $\gamma^{\prime}$ is also an arc of $\Gamma$
connecting two non-adjacent corners of $Q$, and we are done.

Suppose now that $\Gamma$ does not have any arcs connecting two
corners of $Q$. If $\gamma$ has both ends at the same point $p$ then
$Q$ can be replaced by a $(n+1)$-gon $Q^{\prime}$ having all sides of $Q$
plus $\gamma$ as its sides, with the number of faces of the net
$\Gamma^{\prime}$ of $Q^{\prime}$ smaller than $N$. There is a mapping $\iota:Q^{\prime}\to Q$ such that any two distinct points of $Q^{\prime}$ map to distinct points
of $Q$, except the two ends of the side $\gamma$ of $Q^{\prime}$ that both
map to $p$. By the inductive hypothesis, there is an arc $\gamma^{\prime}$
of $\Gamma^{\prime}$ connecting two non-adjacent corners $p^{\prime}$ and $q^{\prime}$ of
$Q^{\prime}$. Then $\iota(\gamma^{\prime})$ is an arc of $\Gamma$ connecting two
(possibly, adjacent) corners of $Q$, a contradiction.

Thus we may suppose that $\gamma$ has two distinct ends $p$ and $q$
on the boundary of $Q$, at least one of them not a corner of $Q$.
Then $\gamma$ partitions $Q$ into two polygons $Q^{\prime}$ and $Q^{\prime\prime}$, with
the number of corners $n^{\prime}$ and $n^{\prime\prime}$ respectively, where $n^{\prime}+n^{\prime\prime}\geq n+3$. If $n>3$ then at least one of $n^{\prime}$ and $n^{\prime\prime}$ is greater than
3. Since both $Q^{\prime}$ and $Q^{\prime\prime}$ have the number of faces of their
nets smaller than $N$, by the inductive hypothesis at least one of
them has an arc $\gamma^{\prime}$ of its net connecting two non-adjacent
corners. Then $\gamma^{\prime}$ is also an arc of $\Gamma$ connecting two
non-adjacent corners of $Q$. This completes the proof.

Corollary 2.3.

If $Q$ is a spherical $n$-gon with $n\geq 3$ satisfying the condition of Theorem
2.2 then an arc $\gamma$ of the net of $Q$ is an
open interval with at least one end at a corner of $Q$.

Proof.

It follows from classification of spherical triangles (see sections
10 and 12 of [Eremenko et al.2016a], section 6 of [Eremenko et al.2016b] and
Figs. 1, 2 and
3a) that an arc of the net of a spherical
triangle with all sides mapped to three circles $C,\,C^{\prime},\,C^{\prime\prime}$ and
all corners to intersection points of those circles must have at
least one end at its corner. Indeed, such a triangle $T$ contains a
primitive triangle $T^{\prime}$ (either one of triangles $T_{\mu}$ or one of
triangles $E_{\mu}$, see Fig. 1) with its apex at
the intersection of two circles, say $C$ and $C^{\prime}$, and its base on
the third circle $C^{\prime\prime}$. Any arc of the net of $T^{\prime}$ connects its apex
with its base. The triangle $T$ is obtained by attaching digons to
the sides of $T^{\prime}$ (see Fig. 3a). An arc of the
net of a digon $D$ either has an end at one of its corners or the
ends on both its sides, but cannot have both ends on one side of
$D$. Thus an arc of $T$ must have at least one end at its corner.
Since the intersection of $\gamma$ with any triangle $T$ of a
triangulation of $Q$ by disjoint arcs of its net connecting its
non-adjacent corners is either a side of $T$ or an arc of the net of
$T$, it must have at least one end at a corner of $T$. But all
corners of $T$ are also corners of $Q$.

Figure 1: Primitive spherical triangles $T_{\mu}$ and $E_{\mu}$

Figure 2: Spherical digons

Figure 3: a Spherical triangle $T_{0}$ with three digons
$D_{1}$ attached to its sides. b Nets of spherical rectangles

As was shown in [Eremenko and Gabrielov2015] (see Proposition 1.1) any
spherical rectangle $Q$ has two opposite sides mapped to the same
circle by its developing map $f$, and two other opposite sides
mapped to distinct circles. Thus there are two types of marked
spherical rectangles: in the first type the images of $L_{2}$ and
$L_{4}$ belong to the same circle, and in the second type the images
of $L_{1}$ and $L_{3}$ belong to the same circle. It is enough to
classify spherical rectangles of the first type, as all rectangles
of the second type can be obtained from those of the first type by
orientation-reversing isometry preserving the marked corner.

Assumption 1.

Unless stated otherwise, all spherical rectangles below are assumed
to be of the first type.

Let $C$ be the circle to which two sides $L_{2}$ and $L_{4}$ of a
spherical rectangle $Q$ are mapped, and let $C^{\prime}$ and $C^{\prime\prime}$ be the
circles to which the sides $L_{1}$ and $L_{3}$ of $Q$ are mapped (see
Fig. 4).

Theorem 2.2 implies that there is an arc of the net
$\Gamma$ of $Q$ connecting two opposite corners of $Q$. Such an arc
must be mapped to the circle $C$, since two opposite corners of $Q$
are mapped to intersection points of $C$ with two distinct circles
other than $C$. This implies that $Q$ cannot have two arcs of
$\Gamma$ connecting two pairs of its opposite corners: such arcs
would have an intersection point inside $Q$, while any two arcs
mapped to the same circle $C$ are disjoint.

Assumption 2.

Unless stated otherwise, we choose the initial corner $a_{0}$ of a
marked spherical rectangle $Q$ so that there is an arc $\gamma$ of
$\Gamma$ connecting the corners $a_{1}$ and $a_{3}$ of $Q$.

Such an arc $\gamma$ partitions $Q$ into two spherical triangles
$T^{\prime}$ and $T^{\prime\prime}$, where $T^{\prime}$ has an integer corner at $a_{3}$ and the
base $L_{1}$ mapped to $C^{\prime}$, while $T^{\prime\prime}$ has an integer corner at
$a_{1}$ and the base $L_{3}$ mapped to $C^{\prime\prime}$ (see Fig. 4). We’ll
show below (see Remark 3.2) that the angles of such
rectangle $Q$ satisfy the inequality $A_{0}+A_{2}+2\leq A_{1}+A_{3}$.

Any rectangle of the first (resp., second) type with an arc of its
net connecting its corners $a_{0}$ and $a_{2}$ can be obtained from a
rectangle of the second (resp, first) type with an arc of its net
connecting its corners $a_{1}$ and $a_{3}$ by choosing $a_{1}$ instead of
$a_{0}$ as an initial corner, and relabeling the corners accordingly.
The angles of such rectangle satisfy the inequality $A_{1}+A_{3}+2\leq A_{0}+A_{2}$. Thus it is enough to classify spherical rectangles
satisfying Assumptions 1 and 2.

Theorem 3.1.

Let $Q$ be a marked spherical rectangle satisfying Assumptions
1 and 2. Then $Q$ is a union of two primitive
triangles $T_{\mu}$ and $T_{\nu}$ having integer angles $\mu+1$ and
$\nu+1$, respectively, digon $D_{2\kappa}$ with the sides mapped to
$C$ having common sides with both $T_{\mu}$ and $T_{\nu}$, digon $D_{i}$
with the sides mapped to $C^{\prime}$ attached to the base of $T_{\mu}$, digon
$D_{l}$ with the sides mapped to $C^{\prime\prime}$ attached to the base of
$T_{\nu}$, digon $D_{m}$ attached to the remaining side of $T_{\mu}$, and
digon $D_{k}$ attached to the remaining side of $T_{\nu}$. The sides of
$D_{k}$ and $D_{m}$ are mapped to $C$.

Here $\mu,\nu,\kappa,i,k,l,m$ are non-negative integers satisfying
$i\mu=l\nu=0$, that is, $i>0$ only if $\mu=0$, $l>0$ only if
$\nu=0$. The value 0 for $i,k,l,m,\kappa$ means there is no digon
attached.

Proof.

Assumptions 1 and 2 imply that an arc $\gamma$
of the net $\Gamma$ of $Q$ connecting its corners $a_{1}$ and $a_{3}$
partitions $Q$ into two spherical triangles $T^{\prime}$ and $T^{\prime\prime}$, where
$T^{\prime}$ has an integer corner with an angle $\mu+1$ at $a_{3}$, and $T^{\prime\prime}$
has an integer corner with an angle $\nu+1$ at $a_{1}$, for some
non-negative integers $\mu$ and $\nu$.

Classification of spherical triangles with one integer corner (see
sections 10 and 12 of [Eremenko et al.2016a]) implies that the triangle $T^{\prime}$
(resp., $T^{\prime\prime}$) is combinatorially equivalent to a primitive triangle
$T_{\mu}$ (resp., $T_{\nu}$) having an angle $\mu+1$ (resp., $\nu+1$) at
its integer corner, with digons attached to its sides (see
Figs. 1, 2, 3a). No digons may be attached to the base
$L_{1}$ of $T^{\prime}$ (resp., the base $L_{3}$ of $T^{\prime\prime}$) if $\mu>0$ (resp.,
$\nu>0$). Each digon $D_{n}$ has equal integer angles $n$ at its two
corners.

The sides of $T_{\mu}$ and $T_{\nu}$ are mapped to $C$ and cannot
contain preimages of intersection points of $C$ with either $C^{\prime}$ or
$C^{\prime\prime}$, other than the corners of $Q$. This implies that the union of
digons attached to the sides of $T_{\mu}$ and $T_{\nu}$ and having
$\gamma$ as their common side is a digon $D_{2\kappa}$ with even
integer angles $2\kappa$ at its two corners.

Thus a net of a spherical rectangle satisfying Assumptions
1 and 2 must have the structure shown
schematically in Fig. 3b. This proves Theorem
3.1.

Remark 3.2.

The angles at the corners $a_{0},a_{1},a_{2},a_{3}$ of a marked spherical
rectangle $Q$ in Theorem 3.1 have the integer parts

respectively. In particular, $A_{0}+A_{2}+2\leq A_{1}+A_{3}$. For a marked
spherical rectangle (of either first or second type) with an arc of
its net connecting its corners $a_{0}$ and $a_{2}$, the integer parts of
its angles satisfy $A_{1}+A_{3}+2\leq A_{0}+A_{2}$.

Remark 3.3.

Theorem 3.1 implies that a spherical rectangle $Q$
satisfying Assumptions 1 and 2 has at least
one short arc $\gamma$ connecting its corners $a_{1}$ and $a_{3}$, and
that all such short arcs are mapped to the same arc $\beta$ of the
circle $C$ with the ends at the intersection points of $C$ with $C^{\prime}$
and $C^{\prime\prime}$.

Definition 3.4.

The angle $\theta\in(0,1)$ between the circles $C^{\prime}$ and $C^{\prime\prime}$ is
defined as $1-\alpha$ where $\alpha$ is the length of any short arc
$\gamma$ of the net of a spherical rectangle $Q$ connecting its
opposite corners, divided by $\pi$. Alternatively, $\alpha$ is the
length of the arc $\beta$ of $C$ to which $\gamma$ is mapped,
divided by $\pi$.

Proof of Proposition 1.2.

We want to show that two marked spherical rectangles $Q$ and $Q^{\prime}$
with equivalent nets $\Gamma$ and $\Gamma^{\prime}$, and the same angle
$\theta$, are congruent, i.e., there is an orientation-preserving
isometry $Q\to Q^{\prime}$ mapping the initial corner $a_{0}$ of $Q$ to the
initial corner $a^{\prime}_{0}$ of $Q^{\prime}$. According to Definition
3.4, the developing map $f:Q\to\mathbf{\overline{C}}$ maps the sides of $Q$
to three great circles $C$, $C^{\prime}$, $C^{\prime\prime}$, such that both $C^{\prime}$ and
$C^{\prime\prime}$ are orthogonal to $C$, so that the sides $L_{2}$ and $L_{4}$ of
$Q$ are mapped to $C$, the sides $L_{1}$ and $L_{3}$ of $Q$ are mapped
to $C^{\prime}$ and $C^{\prime\prime}$, respectively, and any short arc of $\Gamma$
connecting the corners $a_{1}$ and $a_{3}$ of $Q$ is mapped to an arc
$\beta$ of $C$ of length $\pi\alpha$ where $\alpha=1-\theta$, the
ends $P$ and $R$ of $\beta$ being the images of $a_{1}$ and $a_{3}$,
respectively. In particular, $P$ and $R$ are intersection points of
$C$ with $C^{\prime}$ and $C^{\prime\prime}$, respectively. Similarly, the developing map
$g:Q^{\prime}\to\mathbf{\overline{C}}$ maps the sides of $Q^{\prime}$ to three great circles $S$,
$S^{\prime}$, $S^{\prime\prime}$, such that both $S^{\prime}$ and $S^{\prime\prime}$ are orthogonal to $S$,
and any short arc $\gamma^{\prime}$ of $\Gamma^{\prime}$ connecting the corners
$a^{\prime}_{1}$ and $a^{\prime}_{3}$ of $Q^{\prime}$ is mapped to an arc $\beta^{\prime}$ of $S$ of
length $\pi\alpha$, the ends $P^{\prime}$ and $R^{\prime}$ of $\beta^{\prime}$ being the
images of $a^{\prime}_{1}$ and $a^{\prime}_{3}$, respectively. Applying if necessary
rotation of $\mathbf{\overline{C}}$, we may assume that $S=C$, $S^{\prime}=C^{\prime}$, $S^{\prime\prime}=C^{\prime\prime}$,
$\beta^{\prime}=\beta$, $P^{\prime}=P$, and $R^{\prime}=R$. Then the equivalence of the nets
$\Gamma$ and $\Gamma^{\prime}$ implies that the developing maps $f$ and $g$
send the corresponding faces, edges and vertices of partitions of
$Q$ and $Q^{\prime}$ to the same triangles, segments and intersection points
of the partition of $\mathbf{\overline{C}}$ by the three circles $C$, $C^{\prime}$ and $C^{\prime\prime}$.
In particular, $a_{0}$ and $a^{\prime}_{0}$ are mapped by $f$ and $g$ to the
same point of $\mathbf{\overline{C}}$.

Theorem 3.5.

For any non-negative integers $\mu,\nu,\kappa,i,k,l,m$ satisfying
$i\mu=l\nu=0$ there is a unique, up to combinatorial equivalence,
net $\Gamma$ of the type described in Theorem 3.1.
For any such net $\Gamma$ and any $\theta\in(0,1)$ there exists a
unique spherical rectangle $Q=Q(\Gamma,\theta)$ having $\Gamma$ as
its net, sides mapped to three circles $C$, $C^{\prime}$, $C^{\prime\prime}$, and a short
arc of length $\pi(1-\theta)$ connecting its corners $a_{1}$ and
$a_{3}$.

Proof.

To define the net $\Gamma$, we start with a digon $D_{2\kappa}$
obtained by combining $\kappa$ copies of digon $D_{2}$ shown in the
middle of the first row of Fig. 2. If $\kappa=0$
then there is no such digon, and we proceed with gluing together
triangles $T_{\mu}$ and $T_{\nu}$ (see Fig. 1) so that
the side of each of these triangles that follows its base in the
counterclockwise cyclic order becomes their common side. If
$\kappa>0$ then triangles $T_{\mu}$ and $T_{\nu}$ are attached to
opposite sides of $D_{2\kappa}$ so that the side of each of these
triangles that follows its base in the counterclockwise cyclic order
becomes its common side with $D_{2\kappa}$. The integer corner of
$T_{\mu}$ (resp., $T_{\nu}$) coincides with a non-integer corner of
$T_{\nu}$ (resp., $T_{\mu}$).

Next, we attach digons $D_{k}$ and $D_{m}$, obtained by combining $k$
and $m$ copies, respectively, of the digon $D_{1}$ shown in the left
side of the first row of Fig. 2, so that any two
adjacent digons have either a common short side or a common long
side, and so that each of the two resulting digons has at least one
short side, as is shown in examples of $D_{2}$ and $D_{3}$ in the first
row of Fig. 2. Then digons $D_{k}$ and $D_{m}$ are
attached to the free sides of $T_{\nu}$ and $T_{\mu}$, respectively. The
free sides of these triangles are preceding their respective bases
in the counterclockwise cyclic order. If $k=0$ (resp., $m=0$) then
no digon $D_{k}$ (resp., $D_{m}$) is attached.

Finally, if $\mu=0$ and $i>0$ (resp., $\nu=0$ and $l>0$) then a
digon $D_{i}$ (resp., $D_{l}$), obtained by combining $i$ and $l$
copies, respectively, of the digon $D_{1}$ shown in the second row of
Fig. 2, is attached to the base of $T_{\mu}$ (resp.,
$T_{\nu}$). Examples of such digons are shown in the second row of
Fig. 2.

If we label the sides of $T_{\mu}$ and $T_{\nu}$ by $C$, the base of
$T_{\mu}$ by $C^{\prime}$ and the base of $T_{\nu}$ by $C^{\prime\prime}$ then all edges of
$\Gamma$ can be uniquely labeled so that the sides of each of its
triangles are labeled by three distinct labels. Consider the
standard sphere $\mathbf{\overline{C}}$ with three great circles $C$, $C^{\prime}$ and $C^{\prime\prime}$
such that $C$ is orthogonal to both $C^{\prime}$ and $C^{\prime\prime}$, and one of the
two complementary angles between $C^{\prime}$ and $C^{\prime\prime}$ is $\theta$, the
other one being $\alpha=1-\theta$. Then there is a mapping of $Q$ to
$\mathbf{\overline{C}}$, locally one-to-one everywhere except at the corners, which is
unique up to a homeomorphism of $Q$ preserving all vertices and
edges of $\Gamma$ and a rotation of the sphere preserving the three
circles, such that any arc of the boundary of $D_{2\kappa}$ (or a
common side of $T_{\mu}$ and $T_{\nu}$ if $\kappa=0$) maps to an arc of
$C$ of length $\pi\alpha$, and each edge of $\Gamma$ maps to an arc
of the circle corresponding to its label. This defines on $Q$ a
metric of the spherical rectangle $Q(\Gamma,\theta)$.

Each admissible set of integers $\mu,\nu,\kappa,i,k,l,m$ in Theorem
3.1 defines a net $\Gamma$ and the corresponding
one-parametric family of marked spherical rectangles
$Q(\Gamma,\theta)$ satisfying Assumptions 1 and
2, parameterized by the angle $\theta$ between the
circles $C^{\prime}$ and $C^{\prime\prime}$ (see Definition 3.4). These two
circles intersect the circle $C$ at the right angle. In
Fig. 5a, b, two
projections of the three circles are shown, and two of the triangles
of the partition of the sphere defined by these circles are shaded.

Let $P$ and $R$ be the images of the corners $a_{1}$ and $a_{3}$,
respectively, of a marked spherical rectangle $Q$, so that the arc
$\beta=PR$ of $C$ is the image of a short arc $\gamma$ of the net
$\Gamma$ of $Q$ connecting $a_{1}$ and $a_{3}$. Then the shaded areas in
Fig. 5a, b contract to
arcs when $\theta\to 0$ and expand to half-disks when $\theta\to 1$.
If all circles remain geodesic, then $C^{\prime}$ and $C^{\prime\prime}$ converge to the
same circle when $\theta\to 0$ and when $\theta\to 1$. However,
applying a linear-fractional transformation depending on $\theta$,
so that the arc $MN$ of $C$ in Fig. 5a, b contracts to a point while the arc $PR$ does
not, we can obtain in the limit $\theta\to 1$ a non-geodesic
configuration shown in two different projections in
Fig. 5c, d, where the
shaded areas are the limits of the shaded areas in
Fig. 5a, b.

Example 4.1.

A net of a spherical rectangle $Q$ with the angles $(1/2,3/2,1/2,3/2)$ considered in [Eremenko and Gabrielov2015] is shown in
Fig. 6 (left). The shaded area corresponds to
preimage of the shaded areas in Fig. 5a, b. The arc connecting $a_{1}$ and $a_{3}$ is mapped
to the arc $PR$ of $C$, the corners $a_{0}$ and $a_{2}$ are mapped to
$M$ and $N$, respectively. When $\theta\to 0$, the sides $L_{2}$ and
$L_{4}$ of $Q$ are contracted to points, while the distance between
them has a positive limit. Thus the modulus of $Q$ has limit 0 as
$\theta\to 0$ (see [Eremenko et al.2016a], Section 15, and [Eremenko and Gabrielov2015]). When
$\theta\to 1$, applying a linear-fractional deformation depending on
$\theta$ to the three-circle configuration, and the corresponding
transformation to $Q$ (this does not change the modulus of $Q$ which
is a conformal invariant) we can get in the limit a non-geodesic
circular rectangle shown in Fig. 6 (right). Thus
the modulus of $Q$ tends to a finite positive value when $\theta\to 1$. Computation in [Eremenko and Gabrielov2015] shows that this value is $K\approx 0.630963$.

Figure 6: Deformation of spherical rectangles with the angles
$\frac{1}{2},\,\frac{3}{2},\,\frac{1}{2},\,\frac{3}{2}$

Figure 7: Deformation of spherical rectangles with the angles
$\frac{3}{2},\,\frac{5}{2},\,\frac{3}{2},\,\frac{5}{2}$

Figure 8: Deformation of spherical rectangles with the angles
$\frac{3}{2},\,\frac{7}{2},\,\frac{3}{2},\,\frac{7}{2}$

Figure 9: Three combinatorially distinct nets of spherical
rectangles with the angles
$\frac{5}{2},\,\frac{7}{2},\,\frac{5}{2},\,\frac{7}{2}$, and their
deformations

Theorem 4.2.

Let $\Gamma$ be one of the nets described in Theorem
3.1, and $\theta\in(0,1)$. Then the modulus of
$Q(\Gamma,\theta)$ tends to 0 when $\theta\to 0$, and to a finite
positive value when $\theta\to 1$.

Proof.

When $\theta\to 0$, none of the arcs connecting $a_{1}$ and $a_{3}$
contract, while the triangle $T_{\mu}$ (resp., $T_{\nu}$) contains
either a short arc of $\Gamma$ or a side $L_{4}$ (resp., $L_{2}$) of $Q$
that maps to either the arc $MR$ or the arc $NP$ of $C$, connecting
its apex $a_{3}$ (resp., $a_{1}$) with a point $p\neq a_{1}$ (resp., $p\neq a_{3}$) on its base. These two arcs contract to points when $\theta\to 0$. Hence the distance between the sides $L_{1}$ and $L_{3}$ of $Q$
tends to 0 when $\theta\to 0$. At the same time, there are no
short arcs of $\Gamma$ having one end on $L_{2}$ and another end on
$L_{4}$, other than those connecting $a_{1}$ and $a_{3}$ which do not
contract as $\theta\to 0$. Thus the distance between the sides $L_{2}$
and $L_{4}$ does not tend to 0 as $\theta\to 0$. This implies that
the modulus of $Q$ tends to 0 as $\theta\to 0$ (see [Eremenko et al.2016a],
Section 15).

When $\theta\to 1$, the short arcs connecting $a_{1}$ and $a_{3}$
contract, thus both the distance between $L_{1}$ and $L_{3}$ and the
distance between $L_{2}$ and $L_{4}$ tend to 0. To understand the
limit of the modulus of $Q$, we apply a linear-fractional
transformation depending on $\theta$ as in Example 4.1 to the
three-circle configuration, so that the short arc $MN$ of $C$
contracts, while the short arc $PR$ does not. In the non-geodesic
limit (see Fig. 5c) the circles $C^{\prime}$ and $C^{\prime\prime}$
become tangent (when their tangency point is mapped to $\infty$ as
in Fig. 5d, they become parallel lines). All
short arcs of $\Gamma$ connecting $a_{1}$ and $a_{3}$ map to $PR$, and
all arcs of $C$ connecting the apex of a triangle $T_{\mu}$ (resp.,
$T_{\nu}$) with a point on its base, map to either $MR$ or $NP$. Since
neither $MR$ nor $NP$ contracts when $\theta\to 1$, and $PR$ does
not contract after the linear-fractional transformation, the
distances between opposite sides of $Q$ do not tend to 0 in the
limit. Thus $Q$ converges to a non-geodesic circular rectangle (see
Figs. 6, 7, 8, 9) and the modulus of $Q$ tends to a finite
positive value.

Remarks on computation of limit moduli $K$ The boundary of
the limit rectangle described in the proof of Theorem 4.2 is
mapped by developing map into three straight lines (see
Fig. 5d). This allows to represent the
developing map by the Schwarz–Christoffel formula. Condition that
the points $P$ and $Q$ are on the same vertical line imposes one
real equation which permits to determine the modulus of the
rectangle $K$. See [Eremenko and Gabrielov2015] where the simplest example is
described in all detail. The number of solutions to this equation is
the number of nets with given angles. To determine which solution
corresponds to which net, we use the evident inequalities between
the moduli of degenerate rectangles (shown in the right of
Figs. 7, 8, and in the bottom of
Fig. 9).

Example 4.3.

Two combinatorially distinct nets of spherical rectangles with the
angles $(3/2,5/2,3/2,5/2)$, and the nets of their non-geodesic
limits when $\theta\to 1$, are shown in Fig. 7a, b. The moduli $K_{a}$ and $K_{b}$ of the limiting
rectangles are $K_{a}\approx 0.5433144$ and $K_{b}\approx 1.193606$,
respectively. Figure 10 shows schematically the areas of
existence of these spherical rectangles (Nets a and b) and their
involution-symmetric rectangles (Nets $\mathrm{a}^{\prime}$ and
$\mathrm{b}^{\prime}$) for different values of the modulus $K$.

Example 4.4.

Three combinatorially distinct nets of spherical rectangles with the
angles $(5/2,7/2,5/2,7/2)$, and the nets of their non-geodesic
limits when $\theta\to 1$, are shown in Fig. 9p, q, r. The moduli $K_{p}$, $K_{q}$,
$K_{r}$ of the limiting rectangles are $K_{p}\approx 0.476966$,
$K_{q}\approx 0.887943$, $K_{r}\approx 1.458956$, respectively.
Figure 11 shows schematically the areas of existence of these
spherical rectangles (Nets p, q, r) and their involution-symmetric
rectangles (Nets $\mathrm{p}^{\prime}$, $\mathrm{q}^{\prime}$, $\mathrm{r}^{\prime}$) for
different values of the modulus $K$.

Example 4.5.

A net of a spherical rectangle with the angles $(3/2,7/2,3/2,7/2)$,
and the net of its non-geodesic limit when $\theta\to 1$, is shown
in Fig. 8. The modulus $K$ of the limiting
rectangle is $K\approx 0.4173$.

Figure 10: Existence of spherical rectangles with the angles
$\frac{3}{2},\,\frac{5}{2},\,\frac{3}{2},\,\frac{5}{2}$

Figure 11: Existence of spherical rectangles with the angles
$\frac{5}{2},\,\frac{7}{2},\,\frac{5}{2},\,\frac{7}{2}$

Remarks and conjectures Each of the three nets p, q, r in
Fig. 11 produces a continuous family where the modulus $K$
can be arbitrarily small. So for sufficiently small $K$ there are at
least three different marked spherical quadrilaterals with the
angles $(5/2,7/2,5/2,7/2)$. Similarly, one can conclude from
Fig. 11 that there are at least two quadrilaterals with
modulus $K\in(K_{\mathrm{p}},1/K_{\mathrm{r}}$ at least three for
$K\in(1/K_{\mathrm{r}},K_{\mathrm{q}})$, at least two for
$K\in(K_{\mathrm{q}},1/K_{\mathrm{q}})$, at least three for
$K\in(1/K_{\mathrm{q}},K_{\mathrm{r}})$, at least two for
$K\in(K){\mathrm{r}},1/K_{\mathrm{p}})$, and at least three for
$K>1/K_{\mathrm{p}}$.

Similar conclusions apply to Fig. 10. Our computations show
that in fact these lower estimates are equalities, in all cases
which we computed. Actually $K$ is a monotone function of $\theta$
in all these cases, but we do not expect this monotonicity to hold
for all angles.

So Theorem 1.5 gives only lower estimates for the number
of quadrilaterals with given angles and modulus, when this modulus
is small or large. This lower estimate is $N$ or $M_{j}$ (see
(1.10), (1.11), (1.12). We conjecture that these lower
estimates are exact, and that we have equality for large and small
moduli. All computed examples confirm this.

We want to answer the following question: Given four non-negative
integers $A_{0},\dots,A_{3}$, how many nets of marked spherical
rectangles with the integer parts $A_{0},\dots,A_{3}$ of the angles at
their corners $a_{0},\dots,a_{3}$ do exist?

It is enough to answer this question for the spherical rectangles of
the first type (satisfying Assumption 1), since all
rectangles of the second type can be obtained then by an involution
preserving the marked corner. Also, we may assume that $A_{0}+A_{2}+2\leq A_{1}+A_{3}$, which is true for the marked spherical rectangles
satisfying Assumption 2 (see Remark 3.2). The
answer for the rectangles with $A_{1}+A_{3}+2\leq A_{0}+A_{2}$, having an arc
connecting corners $a_{0}$ and $a_{2}$, can be obtained then by a
different choice of the initial corner.

We start with listing operations on the nets which do not change the
angles of a spherical rectangle $Q$. Assuming notation of Theorem
3.1 and Remark 3.2, we have expressions
(3.2) for the angles of $Q$.

Lemma 5.1.

If
$\kappa>0$
and
$\min(i,l)=0$
then there is a unique
operation among Operations I-V that is applicable to
$Q$
and results
in a rectangle with the same angles as
$Q$
, with
$\kappa$
reduced by
1
.

(b)

If
$\min(i,l)=0$
then at most one operation among inverses
to Operations I-V is applicable to
$Q$
. If
$\min(i,l)>0$
then
neither Operations I-V nor their inverses are applicable to
$Q$
.

(c)

If
$\min(i,l)>0$
, then iteration of Operation VI applied
to
$Q$
results in a rectangle with the same angles as
$Q$
, same
$\kappa,\,\mu,\,\nu$
, and
$\min(i,l)=0$
.

Proof.

We start with a proof of (a), assuming $\kappa>0$. If $i=l=0$ then
Operation V is applicable. If $i=1,\;l=0$ then $\mu=0$ and
Operation III is applicable. If $i\geq 2,\;l=0$ then $\mu=0$ and
Operation I is applicable. If $i=0,\;l=1$ then $\nu=0$ and
Operation IV is applicable. If $i=0,\;l\geq 2$ then $\nu=0$ and
Operation II is applicable. It is easy to check that only one of the
operations I–V is applicable in each of these cases.

To prove (b) note first that inverses to Operations I-V are only
possible when $\min(i,l)=0$. Next, for given $\mu$ and $\nu$,
conditions on $\mu$ and $\nu$ for applicability of the operations
inverse to Operations I-V may hold for at most one of these
operations.

If $\min(i,l)>0$ then Operation VI reduces $i,\,l$ and $\min(i,l)$
by 1, and does not change $\kappa,\,\mu,\,\nu$, which proves (c).

Corollary 5.2.

For given $A_{0},\dots,A_{3}$, the set of values of $\kappa$ that may
appear in the nets of marked spherical rectangles of the first type
with $\delta\geq 1$ is either empty (in which case spherical
rectangles with such angles do not exist) or an interval
$[0,\kappa_{\rm max}]$, for some integer $\kappa_{\rm max}\geq 0$
depending on $A_{0},\dots,A_{3}$. In the latter case, there are exactly
$\kappa_{\rm max}+1$ combinatorially distinct nets of marked
spherical rectangles of the first type with given $A_{0},\dots,A_{3}$
and $\min(i,l)=0$.

Lemma 5.3.

Let $Q$ be a marked spherical rectangle with $\min(i,l)=0$
satisfying Assumptions 1 and 2. The net of $Q$
cannot be obtained from a net of some other spherical rectangle by
one of the operations inverse to Operations I-V if and only if one
of the following twelve conditions is satisfied:

(a)

$\mu=\nu=0$
;

(b)

$\mu=0,\;\nu=1$
;

(c)

$\mu=1,\;\nu=0$
;

(d)

$\mu=\nu=1$
;

(e)

$\mu=0,\;\nu=2$
;

(f)

$\mu=2,\;\nu=0$
;

(g)

$\mu=0,\;\nu=3$
;

(h)

$\mu=3,\;\nu=0$
;

(i)

$\mu=1,\;\nu\geq 3,\;m=0$
;

(j)

$\mu\geq 3,\;\nu=1,\;k=0$
;

(k)

$\mu=0,\;\nu\geq 4,\;m\leq 1$
;

(l)

$\mu\geq 4,\;\nu=0,\;k\leq 1$
.

For given $A_{0},\dots,A_{3}$ with $A_{1}+A_{3}\geq A_{0}+A_{2}+2$, a net of a
marked spherical rectangle with $\min(i,l)=0$ satisfying Assumptions
1 and 2 may satisfy at most one of these
conditions.

At most one (up to combinatorial equivalence) net of a marked
spherical rectangle $Q$ satisfying Assumptions 1 and
2, with given $A_{0},\dots,A_{3}$ and $\min(i,l)=0$, may
satisfy any of these conditions. The value of $\kappa$ for such
rectangle $Q$ is

Proof.

One can easily check case by case that none of the operations
inverse to Operations I–V can be applied if and only if one of the
conditions (a)–(l) is satisfied. Note that it is enough to assume
$\mu\leq\nu$, and to check that with this assumption none of the
operations inverse to Operations I,III,V can be applied if and only
if one of the conditions (a), (b), (d), (e), (g), (i), (k) is satisfied.
The case $\mu\geq\nu$ follows by rotation of the net exchanging $a_{0}$
with $a_{2}$, $a_{1}$ with $a_{3}$, $\mu$ with $\nu$, $i$ with $l$, and
$k$ with $m$.

Let now $Q$ be a marked spherical rectangle satisfying Assumptions
1 and 2, with given $A_{0},\dots,A_{3}$,
$A_{1}+A_{3}\geq A_{0}+A_{2}+2$, $\min(i,l)=0$, and $\mu\leq\nu$.

If $Q$ satisfies (a) then $\delta=2\kappa+1$ is an odd integer,
$A_{1}\geq\delta$, $A_{3}\geq\delta$. If, in addition, $l=0$ then $A_{2}=k$
and $A_{3}=\delta+m$, thus
$i=A_{0}-m=A_{0}-A_{3}+\delta=(A_{0}+A_{1}-A_{2}-A_{3})/2$, and the net of $Q$ is
completely determined by its angles. Such a net exists when $i\geq 0$
thus $A_{0}+A_{1}\geq A_{2}+A_{3}$. The case when $i=0$ is treated similarly,
and a net with $i=0$ satisfying (a) exists when $A_{0}+A_{1}\leq A_{2}+A_{3}$. The net with $i=l=0$ satisfying (a) exists when
$A_{0}+A_{1}=A_{2}+A_{3}$.

If $Q$ satisfies (d) then $\delta=A_{3}-A_{0}=A_{1}-A_{2}=2\kappa+2$ is a
positive even integer, $i=l=0$. Then $A_{0}=m$, $A_{2}=k$, and the net
is completely determined by its angles. It exists when
$A_{0}+A_{1}=A_{2}+A_{3}$.

If $Q$ satisfies (e) then $\delta=2\kappa+2$ is a positive even
integer, $A_{1}\geq\delta+1$, $A_{3}\geq\delta-1$, $l=0$, $A_{0}=i+m$,
$A_{2}=k$, $A_{1}=i+k+\delta+1$ thus
$i=A_{1}-A_{2}-\delta-1=(A_{0}+A_{1}-A_{2}-A_{3}-2)/2$, $m=A_{3}-\delta+1$, and
the net is completely determined by its angles. It exists when $i\geq 0$ thus $A_{0}+A_{1}\geq A_{2}+A_{3}+2$. Similarly, if $Q$ satisfies (f) then
$\delta=2\kappa+2$, and its net is completely determined by its
angles. It exists when $A_{1}\geq\delta-1$, $A_{3}\geq\delta+1$,
$A_{2}+A_{3}\geq A_{0}+A_{1}+2$.

Proof.

It follows from Corollary 5.2 and Lemma 5.3 that
existence of a rectangle $Q$ satisfying conditions of Lemma
5.4 implies that the number in (5.2)
is non-negative, which is equivalent to (5.3), and that
the number of combinatorially distinct nets of such rectangles in
that case is the number in (5.2) plus 1, which is
the number in (5.4). Thus we have only to prove
that for any $A_{0},\dots,A_{3}$ satisfying (5.3) there
exists a rectangle $Q$ satisfying conditions of Lemma
5.4.

It follows from Corollary 5.2 that, if a rectangle with
given $A_{0},\dots,A_{3}$ satisfying conditions of Lemma
5.4 exists, there exists also a rectangle with
$\kappa=0$ with the same angles satisfying the same conditions. We
are going to construct a net of such a rectangle for any
$A_{0},\dots,A_{3}$ satisfying (5.3).

There are three possible cases: (i) $A_{1}>A_{2}$ and $A_{3}>A_{0}$; (ii)
$A_{1}>A_{2}$ and $A_{3}\leq A_{0}$; (iii) $A_{1}\leq A_{2}$ and $A_{3}>A_{0}$. Note
that $A_{1}\leq A_{2}$ and $A_{3}\leq A_{0}$ is not possible because
$\delta=(A_{1}+A_{3}-A_{0}-A_{2})/2\geq 1$.\listatt

Definition 5.5.

A marked spherical rectangle is special if $\delta$ is an odd
integer and either $A_{1}\geq\delta>0$ and $A_{3}\geq\delta>0$ or $A_{0}\geq-\delta>0$ and $A_{2}\geq-\delta>0$.

Lemma 5.6.

A marked spherical rectangle $Q$ satisfying Assumptions 1
and 2 with given $A_{0},\dots,A_{3}$ is special if an only if
there exists a rectangle with the same angles whose net has
$\mu=\nu=0$. For given $A_{0},\dots,A_{3}$ satisfying conditions of
Definition 5.5 with $\delta>0$, there are

$\displaystyle\min(A_{0},A_{1}-\delta,A_{2},A_{3}-\delta)$

(5.5)

special rectangles satisfying Assumptions 1 and
2, with $\mu=\nu=0$ and $\min(i,l)>0$.

Proof.

If $\min(i,l)>0$ for the net of $Q$ then $\mu=\nu=0$. From Lemma
5.1 (c), iteration of Operation VI applied to $Q$ results in
a unique rectangle $Q_{0}$ with the same angles as $Q$, $\mu=\nu=0$
and $\min(i,l)=0$. Thus $Q_{0}$ satisfies Lemma 5.3 (a). If
$\min(i,l)=0$ for the net of $Q$ then, from Corollary 5.2,
there is a unique rectangle with the same angles as $Q$ satisfying
one of the conditions (a)-(l) of Lemma 5.3. One can
easily check that $\delta$ is an odd integer, $A_{1}\geq\delta$ and
$A_{3}\geq\delta$ only in case (a) of Lemma 5.3. Conversely,
a rectangle satisfying condition (a) of Lemma 5.3 is
clearly special, thus any rectangle $Q$ with the same angles is also
special. Finally, the number in (5.5) is obtained
by counting distinct rectangles that can be obtained from a
rectangle with $\mu=\nu=0$ by iterating Operation VI and its inverse
(compare with Lemma 11.2 in [Eremenko et al.2016a]).

Theorem 5.7.

Let $Q$ be a marked spherical rectangle satisfying Assumptions
1 and 2, with given $A_{0},\dots,A_{3}$. If $Q$ is
not special then there are (5.4) combinatorially
distinct nets of marked spherical rectangles satisfying Assumptions
1 and 2 with the same angles as $Q$. If $Q$ is
special then the number in (5.4) is
$(1+\delta)/2$, and there are additionally (5.5)
nets of marked spherical rectangles satisfying Assumptions
1 and 2 with the same angles as $Q$, thus the
total number of combinatorially distinct nets is

Proof.

This follows from Lemmas 5.4 and 5.6.

Proof of Theorems 1.3 and 1.5..

Lemmas 5.4 and 5.6 imply that a marked
spherical rectangle satisfying Assumptions 1 and
2 exists if and only if (5.3) holds. Since
this condition is symmetric with respect to $A_{1}$ and $A_{3}$, it
remains true for spherical rectangles of the second type satisfying
$A_{0}+A_{2}<A_{1}+A_{3}$, as any such rectangle can be obtained from a
rectangle satisfying Assumptions 1 and 2 by a
reflection preserving $a_{0}$ and $a_{2}$, exchanging $a_{1}$ and $a_{3}$.

If $Q$ is a marked spherical rectangle of either first of second
type satisfying $A_{0}+A_{2}>A_{1}+A_{3}$, replacing $a_{0}$ by $a_{1}$ as an
initial corner, and relabeling the corners accordingly, results in a
marked spherical rectangle $Q^{\prime}$ of either second or first type, with
the integer parts of the angles
$(A^{\prime}_{0},A^{\prime}_{1},A^{\prime}_{2},A^{\prime}_{3})=(A_{1},A_{2},A_{
3},A_{0})$ and
$\delta^{\prime}=(A^{\prime}_{1}+A^{\prime}_{3}-A^{\prime}_{0}-A^{\prime}_{2})/
2=-\delta$. Applying the above
arguments to $Q^{\prime}$ we see that a marked spherical rectangle $Q$ with
$A_{0}+A_{2}>A_{1}+A_{3}$ exists if and only if

$\displaystyle\min\left(A_{0},A_{2},-\delta\right)\geq 1,$

(5.7)

Combining (5.3) and (5.7) we get the
statement of Theorem 1.3. The statement of Theorem
1.5 follows from (5.4) and
(5.6) applied to either $Q$ or $Q^{\prime}$ in a similar
way.

Lemma 5.8.

Let $Q$ be a marked spherical rectangle with the angles satisfying
$A_{0}=A_{2},\;A_{1}=A_{3}$. Then there is an orientation-preserving
isometry $\rho:Q\to Q$ such that $\rho(a_{0})=a_{2}$ and
$\rho(a_{1})=a_{3}$.

Proof.

Due to Proposition 1.2, it is enough to prove that the
net $\Gamma$ of $Q$ is symmetric with respect to a transformation
exchanging $a_{0}$ with $a_{2}$ and $a_{1}$ with $a_{3}$, and that this
symmetry of $\Gamma$ maps any short arc of $\Gamma$ connecting $a_{1}$
with $a_{3}$ is mapped to a (possibly, different) short arc connecting
$a_{1}$ with $a_{3}$. To show this, we have only to check (assuming that
$Q$ satisfies Assumptions 1 and 2) that
$\mu=\nu$, $i=l$, and $k=m$ in the notations of Theorem
3.1.

Suppose first that $\mu=\nu=0$. Then $A_{0}=A_{2}$ implies $i+m=k+l$,
and $A_{1}=A_{3}$ implies $i+k=m+l$ (see (3.2) in Remark
3.2). Adding up these two equalities yields $i=l$, and
subtracting them yields $k=m$.

If $\mu>0$ and $\nu>0$ then $i=l=0$, thus $A_{0}=m$, $A_{2}=k$,
$A_{1}=k+1+2\kappa+\nu$, and $A_{3}=m+1+2\kappa+\mu$. Since $A_{0}=A_{2}$,
we have $k=m$, then $A_{1}=A_{3}$ yields $\mu=\nu$.

If $\mu>0$ but $\nu=0$ then $i=0$, thus $A_{0}=m$, $A_{1}=k+1+2\kappa$,
$A_{2}=k+l$, and $A_{3}=l+m+1+2\kappa+\mu$. Since $A_{0}=A_{2}$, we have
$m=k+l$, thus $A_{3}=k+2l+1+2\kappa+\mu>A_{1}$, a contradiction.
Similarly, $\mu=0$ and $\nu>0$ is not possible. This completes the
proof.

Theorem 5.9.

A spherical rectangle $Q$ with the angles at two of its opposite
corners equal $\alpha$, and the angles at two other opposite corners
equal $\beta$, exists if and only if $|\beta-\alpha|\geq 1$. If
$\beta-\alpha$ is even then there are $|\beta-\alpha|$ non-isometric
spherical rectangles with these angles, $|\beta-\alpha|/2$ of them
satisfying Assumption 1. If $\beta-\alpha$ is odd then
there are $\alpha+\beta$ non-isometric spherical rectangles with
these angles, $(\alpha+\beta)/2$ of them satisfying Assumption
1.

This follows from Lemma 5.8 and Theorems 1.3
and 1.5.

Example 5.10.

The net in Fig. 6 is special, with $A_{0}=A_{2}=0$,
$A_{1}=A_{3}=1$, $\delta=1$. According to (5.6) and
Theorem 5.9, there is a unique net of a marked
spherical rectangle of the first type with these angles.

The two nets in Fig. 7 are special with
$A_{0}=A_{2}=1$, $A_{1}=A_{3}=2$, $\delta=1$. According to
(5.6) and Theorem 5.9, there are
two nets of marked spherical rectangles of the first type with these
angles.

The net in Fig. 8 is not special, with
$A_{0}=A_{2}=1$, $A_{1}=A_{3}=3$, $\delta=2$. According to
(5.4) and Theorem 5.9, there is a
unique net of a marked spherical rectangle of the first type with
these angles.

The three nets in Fig. 9 are special with
$A_{0}=A_{2}=2$, $A_{1}=A_{3}=3$, $\delta=1$. According to
(5.6) and Theorem 5.9, there are
three nets of marked spherical rectangles of the first type with
these angles.